**4. Simulation Study**

To verify the actual control effect of the improved LADRC , a vector control model of a three-phase asynchronous motor based on rotor flux linkage orientation was established based on MATLAB and Simulink simulation software (Development by MathWorks, Natick, MA, USA, and agen<sup>t</sup> of MathWorks Software (Beijing) Co., Ltd.). The motor parameters are shown in Table A1. The current loop controller adopted the improved LADRC control, and the outer loop adopted Proportional integral controller (PI) control. The control effect of the controller was verified by simulating the motor speed, the three-phase stator current, the electromagnetic torque, and the sudden load when the motor was started without load at different speeds.

### *4.1. Dynamic Performance of the Controller for an Induction Motor at Di*ff*erent Given Speeds*

### 4.1.1. Dynamic Performance of the Controller Given a Low Speed of the Motor

Figure 14a shows the simulation diagram of the no-load starting process of the asynchronous motor with a given reference speed of 200 rev/min. The red curve represents the improved LADRC, while the blue curve represents the traditional LADRC. It can be seen from Figure 14 that the traditional LADRC could reach the given value near the motor speed at 0.1 s, while the improved LADRC could reach the given value near the motor speed at 0.07 s. Therefore, the controller effect of the improved LADRC was better than that of the traditional LADRC.

**Figure 14.** Low-speed simulation of an asynchronous motor. LADRC: Linear ADRC.

Figure 14b is a simulation diagram of the electromagnetic torque of the induction motor during no-load starting. The corresponding speed of the electromagnetic torque of the motor was greater when the improved LADRC was adopted. The electromagnetic torque of the motor when the improved LADRC was adopted was larger than that when the traditional LADRC was adopted at the same time point; therefore, the electromagnetic torque of the motor could be restored to zero in a shorter time.

Figure 14c is the simulation diagram of the three-phase stator current when the asynchronous motor started up without a load under the action of the traditional LADRC. After the improved LADRC was adopted, the three-phase stator current reached the steady-state again near 0.09 s, and the motor speed reached the given value. Figure 14d is the simulation diagram of the three-phase stator when the asynchronous motor started up without a load under the action of the improved LADRC. After the improved LADRC was adopted, the three-phase stator current reached the steady-state again near 0.07 s. It can be seen from the simulation diagram of the three-phase stator under the action of two controllers that the dynamic performance of the improved linear observer was better.

The improved linear expansion observer can better estimate the total disturbance in the system and realize the decoupling between the excitation subsystem and the torque subsystem. Therefore, the motor controlled by the improved Linear Active Disturbance Rejection Controller at the same time in Figure 14b can obtain a greater electromagnetic torque. The key to motor speed regulation is the adjustment of electromagnetic torque. At the same time in Figure 14b, the electromagnetic torque obtained by the motor under the improved linear auto-disturbance control is larger than that under the traditional linear auto-disturbance controller, so the improvement in Figure 14a Under the Linear Active Disturbance Rejection Controller, the motor speed can reach the given value of a shorter time.

The load in the equivalent circuit of the motor is purely resistive and the magnitude of the equivalent resistance is related to the slip rate. The slip rate is 1 when the motor is started without load. At this time, the total impedance of the system is small and the current on the stator side is large. When the motor reaches a given speed, the slip ratio is less than 1 and the total impedance of the system becomes larger. Therefore, the three-phase stator currents in Figure 14c,d become smaller when the motor speed reaches a given value.

### 4.1.2. Dynamic Performance of the Controller Given a High Speed of the Motor

Figure 15a is the speed simulation diagram of the asynchronous motor at a given reference speed of 800 rev/min, in which the blue curve is the simulation diagram under the action of the traditional LADRC, and the red curve is that under the action of the improved LADRC. It can be seen from the figure that the traditional LADRC reached the given value near the motor speed at 0.23 s, while the improved LADRC reaches the given value near the motor speed at 0.18 s, and exhibited a better control effect than the traditional LADRC.

Figure 15c is the simulation diagram of the three-phase stator when the asynchronous motor started up without a load under the action of the traditional LADRC. After the improved LADRC was adopted, the three-phase stator current reached the steady-state again near 0.17 s, when the motor speed reached the given value. Figure 15d is the simulation diagram of the three-phase stator when the asynchronous motor started up without a load under the action of the improved LADRC. After the improved LADRC was adopted, the three-phase stator current reached the steady-state again near 0.16 s. According to Figure 15c,d, the dynamic performance of the improved linear observer was better.

**Figure 15.** High-speed simulation of the asynchronous motor.

### *4.2. Steady State Error of Asynchronous Motor Controller at Di*ff*erent Given Speeds*

Figure 16a is the local amplification of Figure 15a. Although the steady-state error of the system under the action of the traditional Linear Active Disturbance Rejection Controller is not large at a given reference speed of 200 r/min, the steady-state error of the asynchronous motor under the action of the improved Linear Active Disturbance Rejection Controller is smaller. Figure 16b is the local amplification of Figure 16a. Compared with the given reference speed of 200 r/min, the steady-state error of the asynchronous motor under the action of the improved Linear Active Disturbance Rejection Control is still better than the traditional Linear Active Disturbance Rejection Controller, which, although, becomes larger under the given reference speed of 200 r/min.

**Figure 16.** Steady-state error of the system at different speeds.

PI control uses error feedback to eliminate the error. When the system output is equal to the given input, the controller output is zero. At this time, the motor cannot maintain the current speed and deviates from the given value, resulting in the steady-state error of motor speed. The coupling between the excitation subsystem and the torque subsystem increases with the increase of the speed, resulting in the steady-state error of the speed of the motor in Figure 16b at high speed higher than that in Figure 16a at low speed.

### *4.3. Immunity Performance of the Asynchronous Motor Controller at Di*ff*erent Given Speeds*

Figure 17a is a simulation diagram of the motor speed during the process of a sudden increase in the mechanical torque by 10 N·<sup>m</sup> in 0.35 s and a subsequent sudden decrease in the mechanical torque by 10 N·<sup>m</sup> in 0.4 s at a given speed of 200 rev/min (returning to the state before the system loading). Figure 17b is a simulation diagram of the motor speed during the process of a sudden increase in the mechanical torque by 10 N·<sup>m</sup> in 0.35 s and subsequent sudden decrease in the mechanical torque by 10 N·<sup>m</sup> in 0.4 s (returning to the state before the system loading) at a given speed of 800 rev/min. From Figure 17a,b, it can be seen that the motor speed controlled by the traditional LADRC fluctuated more after the mechanical load of the motor suddenly increased; therefore, the immunity of the improved LADRC was better than that of the traditional LADRC.

**Figure 17.** Sudden load increase and decrease of the motor at different speeds.

The coupling between the excitation subsystem and the torque subsystem increases to the increase in speed, resulting in the steady-state error of the motor in Figure 17b at a high speed greater than that in Figure 17a at low speed. The improved linear expansion observer can estimate the total disturbance more accurately, reduce the coupling between the two subsystems to a certain extent, and improve the control of electromagnetic torque. Since the improved Linear Active Disturbance Rejection Controller is applied to the current inner loop decoupling and the outer loop still adopts PI control, the improvement of load torque and the steady-state error is not obvious.
